# Finite Element Method Mcqs

Q:

A) Transverse shear strains in thick plates present computational difficulties | B) Transverse shear strains in thin plates present computational efficiency |

C) For thick plates, the element stiffness matrix yields erroneous results for the generalized displacements | D) For thin plates, the element stiffness matrix becomes stiff and yields erroneous results |

Answer & Explanation
Answer: D) For thin plates, the element stiffness matrix becomes stiff and yields erroneous results

Explanation: The transverse shear strains in the element equations of Shear Deformation Theory cause computational difficulties when the side-to-thickness ratio of the plate is large. Shear locking is observed when the transverse shear strains in thin plates are negligible, and consequently, the element stiffness matrix becomes stiff and yields erroneous results for the generalized displacements.

Explanation: The transverse shear strains in the element equations of Shear Deformation Theory cause computational difficulties when the side-to-thickness ratio of the plate is large. Shear locking is observed when the transverse shear strains in thin plates are negligible, and consequently, the element stiffness matrix becomes stiff and yields erroneous results for the generalized displacements.

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Q:

A) If the plate is thick | B) If the side to thickness ratio of the plate is large |

C) If the side to thickness ratio of the plate is small | D) If higher-order finite elements are used |

Answer & Explanation
Answer: B) If the side to thickness ratio of the plate is large

Explanation: The transverse shear strains in the element equations of Shear Deformation Theory present computational difficulties when the side-to-thickness ratio of the plateis large (say 50, i.e., when the plate becomes thin). For thin plates, the transverse shear strains are negligible, and consequently, the element stiffness matrix becomes stiff and yields erroneous results for the generalized displacements. This phenomenon is known as shear locking.

Explanation: The transverse shear strains in the element equations of Shear Deformation Theory present computational difficulties when the side-to-thickness ratio of the plateis large (say 50, i.e., when the plate becomes thin). For thin plates, the transverse shear strains are negligible, and consequently, the element stiffness matrix becomes stiff and yields erroneous results for the generalized displacements. This phenomenon is known as shear locking.

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Q:

A) The plane elasticity equations govern the transverse deflections | B) The transverse deflections are coupled with in-plane displacements |

C) The in-plane displacements are zero in the absence of in-plane forces | D) The transverse deflections are zero in the absence of in-plane forces |

Answer & Explanation
Answer: C) The in-plane displacements are zero in the absence of in-plane forces

Explanation: For a linear theory based on infinitesimal strains and orthotropic material properties, the in-plane displacements (ux, uy) are uncoupled from the transverse deflection uz=w. The plane elasticity equations govern the in-plane displacements (ux, uy). The in-plane displacements are zero if there are no in-plane forces and hence, we discuss only the equations governing the bending deformation and the associated finite element models.

Explanation: For a linear theory based on infinitesimal strains and orthotropic material properties, the in-plane displacements (ux, uy) are uncoupled from the transverse deflection uz=w. The plane elasticity equations govern the in-plane displacements (ux, uy). The in-plane displacements are zero if there are no in-plane forces and hence, we discuss only the equations governing the bending deformation and the associated finite element models.

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Q:

A) A straight-line perpendicular to the plane of the plate is inextensible | B) A straight line perpendicular to the plane of the plate remains straight |

C) A straight line perpendicular to the plane of the plate rotates such that it remains perpendicular to the tangent to the deformed surface | D) A straight line perpendicular to the plane of the plate rotates |

Answer & Explanation
Answer: C) A straight line perpendicular to the plane of the plate rotates such that it remains perpendicular to the tangent to the deformed surface

Explanation: In the SDT, we relax the normality assumption of CPT, i.e., transverse normal may rotate without remaining perpendicular to the mid-plane. The Classical Plate Theory is based on the assumption that a straight line perpendicular to the plane of the plate is (1) inextensible, (2) remains straight, and (3) rotates such that it remains perpendicular to the tangent to the deformed surface.

Explanation: In the SDT, we relax the normality assumption of CPT, i.e., transverse normal may rotate without remaining perpendicular to the mid-plane. The Classical Plate Theory is based on the assumption that a straight line perpendicular to the plane of the plate is (1) inextensible, (2) remains straight, and (3) rotates such that it remains perpendicular to the tangent to the deformed surface.

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Q:

A) It is also called Kirchhoff plate theory | B) It is an extension of Euler-Bernoulli beam theory from one dimension to two dimensions |

C) It does not involve Timoshenko beam theory | D) It is often known as Hencky-Mindlin plate theory |

Answer & Explanation
Answer: D) It is often known as Hencky-Mindlin plate theory

Explanation: The two most commonly used displacement-based plate theories are the Classical Plate Theory (CPT) and first-order Shear Deformation Theory (SDT). CPT is an extension of the Euler-Bernoulli beam theory from one dimension to two dimensions and is also known as the Kirchhoff plate theory. Shear Deformation Theory is an extension of the Timoshenko beam theory and it is often called the Hencky-Mindlin plate theory.

Explanation: The two most commonly used displacement-based plate theories are the Classical Plate Theory (CPT) and first-order Shear Deformation Theory (SDT). CPT is an extension of the Euler-Bernoulli beam theory from one dimension to two dimensions and is also known as the Kirchhoff plate theory. Shear Deformation Theory is an extension of the Timoshenko beam theory and it is often called the Hencky-Mindlin plate theory.

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Q:

A) The transverse deflection w only | B) The transverse deflection w and the normal derivative of w |

C) The transverse deflection w and the angles of rotation of the transverse normal about in-plane axes | D) The angles of rotation of the transverse normal about in-plane axes only |

Answer & Explanation
Answer: C) The transverse deflection w and the angles of rotation of the transverse normal about in-plane axes

Explanation: An examination of the boundary terms in the weak form of Shear Deformation Theory suggests that the essential boundary conditions involve specifying the transverse deflection w and the angles of rotation of the transverse normal about in-plane axes (φx, φy), which constitute the primary variables of the problem (like in the Timoshenko beam model). Hence, the finite element interpolation of w must be such that w, (φx and φy are continuous across the inter-element boundaries in SDT elements.

Explanation: An examination of the boundary terms in the weak form of Shear Deformation Theory suggests that the essential boundary conditions involve specifying the transverse deflection w and the angles of rotation of the transverse normal about in-plane axes (φx, φy), which constitute the primary variables of the problem (like in the Timoshenko beam model). Hence, the finite element interpolation of w must be such that w, (φx and φy are continuous across the inter-element boundaries in SDT elements.

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Q:

A) True | B) False |

Answer & Explanation
Answer: B) False

Explanation: For a linear theory based on infinitesimal strains and orthotropic material properties, the in-plane displacements(ux, uy) are uncoupled from the transverse deflection uz=w. The plane elasticity equations govern the in-plane displacements (ux, uy). The in-plane displacements are zeroin the absence of in-plane forces, andhence, we discuss only the equations governing the bending deformation.

Explanation: For a linear theory based on infinitesimal strains and orthotropic material properties, the in-plane displacements(ux, uy) are uncoupled from the transverse deflection uz=w. The plane elasticity equations govern the in-plane displacements (ux, uy). The in-plane displacements are zeroin the absence of in-plane forces, andhence, we discuss only the equations governing the bending deformation.

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Q:

A) Classical Plate Theory | B) Hencky-Mindlin plate theory |

C) Kirchhoff plate theory | D) Shell theory |

Answer & Explanation
Answer: B) Hencky-Mindlin plate theory

Explanation: The two most commonly used displacement-based plate theories are the Classical Plate Theory (CPT) and first-order Shear Deformation Theory (SDT). CPT is an extension of the Euler-Bernoulli beam theory from one dimension to two dimensions and is also known as the Kirchhoff plate theory. Shear Deformation Theory is an extension of the Timoshenko beam theory and it is often called the Hencky-Mindlin plate theory.

Explanation: The two most commonly used displacement-based plate theories are the Classical Plate Theory (CPT) and first-order Shear Deformation Theory (SDT). CPT is an extension of the Euler-Bernoulli beam theory from one dimension to two dimensions and is also known as the Kirchhoff plate theory. Shear Deformation Theory is an extension of the Timoshenko beam theory and it is often called the Hencky-Mindlin plate theory.

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